Asymptotics of heat kernels on non-euclidean spaces and spectrum

非欧空间和谱上热核的渐近

• 批准号: 15540189
• 负责人: ICHIHARA Kanji
• 金额: $ 2.3万
• 依托单位: Kansai University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540189/
• 关键词: radial random walk; homogenous tree; large deviation; rate function; Brownian motion; spectrum; curvature; principal eigenfunction; ロバチェフスキー平面; 基本領域; 離散群; 保形関数; 測地流; stable foliation; harmonic measure; 双曲的リーマン多様体; 結晶格子; 可逆的マルコフ連鎖; spheri

An infinite homogenous tree is a typical example of non-euclidean discrete spaces. We have established Donsker-Varadhan's type large deviation for the pinned motions of a radial random walk on the above tree. It has been shown that the corresponding rate function is related to a new Markov chain defined through harmonic transform based on a positive principal eigenfunction for the generator associated with the original random walk. Note that the principal eigenfunction depends only on the structure of the tree. Secondly, there have discussed the same problems for Brownian motions on a class of hyperbolic Riemannian manifolds whose sectional curvature diverges to -∞ at infinity. We have succeeded in showing the uniform large deviation principle for this case. Namely the upper bound is proven to be valid for any closed subset. For the manifold the bottom of the spectrum of the negative Laplacian is discrete and the associated principal eigenfunction decays faster than in an exponential order. Thirdly, the explosion problem for a continuous time, reversible Markov chains on a countably infinite set has been discussed from the viewpoint of Dirichlet space.

无限的同质树是非欧对人离散空间的典型例子。我们已经建立了Donsker-Varadhan的类型大型出发，以固定在上面的树上径向随机行走的动作。已经表明，相应的速率函数与基于与原始随机步行相关的发电机的正主征函数定义的新马尔可夫链有关。请注意，主征函数仅取决于树的结构。其次，在一类双曲riemannian歧管上讨论了布朗尼运动的问题，它们的截面曲率在无穷大时偏向-∞。我们已经成功地展示了这种情况的统一大偏差原则。即，上限被证明对任何闭合子集有效。对于歧管，负拉普拉斯的频谱的底部是离散的，相关的主征性衰减更快，第三，连续时间的爆炸问题，从迪里奇莱特空间的角度讨论了乡村无限集中的可逆马尔可夫链。